Federal State Educational Institution of Higher Professional Education “Lobachevsky State University of Nizhni Novgorod’'
A manuscript
AUGELLO Giuseppe
TRANSIENT DYNAMICS OF SHORT AND LONG JOSEPHSON JUNCTIONS
01.04.03 — Radiophysics
DISSERTATION for the degree Candidate of Physical and Mathematical Sciences
Supervisors: Ph.D., Professor Spagnolo Bernardo (Italy), Candidate of Physical and Mathematical Sciences, Associate Professor Dubkov Alexander Alexandrovich
Nizhni Novgorod - 2012
Contents
1 Introduction 8
1.1 Transient Dynamics mid Josephson .1 unctions.................. 8
1.2 Brief Review on Superconductivity and Josephson Junction . . 10
J.3 Quantum computing and Josephson Junction».....................12
1.4 Effects of noise in Josephson Junctions...................................14
1.4.1 Short and Long Josephson Junctions.............................. 14
112 Noise Induced Effects: resonant activation and noise
enhanced stability...................................... 15
1.4.3 Thermal Noise in Short. Joscplison Junctions.................... 18
1.4/1 Low Frequency Noise in Josephson Junctions...................... 24
1.4.5 Non-Gaussian Noise in Josephson junctions .......................27
1.4.6 Thermal Noise in Long Josephson Junctions........................29
1.5 Colored and Non-Gaussian Noise in Josephson Junctions: overview
of t he project................................................33
2 Short Josephson Junctions 35
2.1 Equivalent Circuit and Mechanical Model.......................35
2.2 Colored Noise.................................................41
2 3 Effects of Colored Noise................................ .42
2.4 Non-Gaussian Noise............................................48
2.5 Effects of uon-Gaussiaii Xoise................................52
2.6 Simultaneous action of non-Gaussimi and thcrimil noise .... 59
3 Long Josephson Junctions G2
3.1 Equivalent Circuit and Mechanical Model.......................62
3.2 Correlated Noise and sine-Gordon Equation.....................06
3 o Effects of Colored Xoise......................................69
3 4 Simultaneous Action of Colored and Thermal White Noise . . 78
4 Conclusions 81
4.1 Main results of this work.....................................81
4.2 Future research projects...................................... 84
Bibliography 86
A Algorithm for the Simulation of Stable Random Variables 93
A.l Stable Variables and Distributions............................ 93
A.2 Simulation Algorithm..........................................93
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CONTENTS
B Algorithm for the Numerical Integration of the sine-Gordon Equation 95
C Publications 99
C.l Papers in ISI Journals..............................................99
C.2 Proceedings.........................................................99
C’.3 Abstracts in international conferences............................100
CONTENTS
Rationale
In recent yours a great attention was paid to the stud> of Josephson jun< lions because of I heir use both as superconducting quantum bits (2j and nunoscalc superconducting quantum interference devices for detecting weak magnetic flux changes [3j. For this reason, they were studied at very low teinpciatuio in devices making use ol ehaige. (lux and phase quints.
In Josephson junctions, working both at high and low Ципрегаїшеь, the environment afhxt.s strongly the behavior of the system. In high temperature superconductors (HTSs) the presence of low-frequency noise, whose intensity is related to the fluctuations in the bias current, temperature and magnetic held, was expci inienUiHv found [9] Also in the low temperature superconductive devices it is very difficult to avoid t.lic influence of environment Ihal constitutes mainly a dccohcLcnec source for the system. In particular, the effects on the coherence time of weak current noise in Josephson vortex qubils (JVQs). composed by on uniform long Josephson junction, were investigated [21
In this framework the study of transient dynamics of Josephson junctions in the presence of noise somces is very interesting for the undcistanding of the interaction between these systems and the environment. In particular, the effects of noise slLongly influence the curreiit-voltage characteristic of Josephson junctions [12].
The behavior of llic cut rent-voltage characteristic of a Josephson junction is strictly i plated to the lifetime of the metastable state of the system. Recently. noise induced effects wcie experimentally observed in undcrdamped Josephson pmrtions [17.18], and the switching of an annular Josephson junction due to thermal activation was analyzed [10].
The goal of the dissertation work is the study of the efforts of different noise sources, such as colored and non-Gaussian noise, on the dynamics ol shoit and long Josephson junctions through the realization of suitable computing codes.
Research methods and authenticity of scientific results. The
study pcrfoimed in the present work is based on the use of well known stochastic processes analysis and computational methods. The validity ol the achievements is granted by the comparison with the results, available in literal me. concerning the effects of white noise on short and long Josephson pirn lions.
Scientific novelty The efforts of white noise on the dynamics of short and long Josephson junctions were studied in detail under experimental, theoretical and simulative approach. The study proposed in the present work concerns the effects of noil-thermal noise sotuccs on these systems and their possible use to improve the performance of the devices
CONTENTS
Theoretical and practical contribution The computational codes structure, developed for the presented study, can be adapted for the investigation of the dynamics of Josephson junctions under different physical condition. In particular, it could be applicable for the realization of a simulative code to analyze sohton diffusion in long .Josephson junctions under tlie influence ol coloicd and non-Gaussian noise souices
Theses for defence:
1. Ill«4 noise induced effects, resonant activation and noise enhanced sta-hilitv allow to find tango of the charm teristic parameters, місії as fro-qucticv of the driving and intensity of noise, in which the metastability of the system can be speeded up or enhanced.
2. The met astabilitv of t he system can be cunt rolled under t he vai iation of pceiilidiities of the noise such as the cut-off frequency of the cot related signal or the probability density function of the noise signal.
3. Knowledge of the behavior of the met as lability of the system undci the simultaneous action of thermal and non-thermal noise is useful to identify the operation range m which the effects ol noil-thermal noise are relevant.
Approbation of the results. The main results of the dissertation work have been presented at the following conferences: International Conference on Complexity. Meloblalnlily пік/ Nonexlensivdy. Satellite Conference of S1’ATPHYS23. Catania, Italy, lth-5fch July 2007; Noise Information and Complexity al Quantum Scale. Ettoie Majorana Centro. Ericc, Italy, 4th-10th November '2007; In! ana bond Conference, on Statistical Physics. 14th-18th .Julv 2008.. Orthodox Academy of Crete Kolympari, Chania, Greece; 22u,/ General Conference of the Condensed Mailer Division of I he European Physical Society, 25th-29th August 200S. ’’La Sapienza " University, Home, Italy: Fourth International Workshop DICE200S. 22lli-26th Seinptember 2008. Gastello Pasquini/Castiglioncello, Italy; 22utl Marian Smolvrhmcsb Symposium on Statistical Physics, Fundamentals und Applications. 12th- 17th September 2009. Zakopane, Poland.
The results of the present work were discussed at the scientific seminars at the Hadiophvsies faculty of Nizhni Novgorod State University (Russia) in June 2008 and 2009 and at Diparttmento di Ftstca e Tecnologie, Relative of Palermo University (Italy) in November 2007, 2008 and 2009.
Author’s publications. The results presented in the dissertation wore published in 6 scientific papers, including 3 papers m IS1 journals [PI. P3. Р4І and 5 proceedings of conferences [PP1. PP4. Al. A5. AG;.
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CONTENTS
Author’s contribution. The present work is the result of the coJlnb-oralion of a research group. The author participated actively to the investigation being the principal investigator in tlie research topic, realization of computational codes and interpretation of t he results.
Structure and amount of the dissertation. The dissertation consists of Introduction, two chapters, Conclusions, Bibliography and three appendixes including the list of author's publications. The whole amount of the dissertation is 10J pages, including 85 pages of main text. 43 figures and 3 tabic.
Acknowledgements
1 wish to thank my tutor, professor Bernardo Spagnolo, for giving me the opport unitv to consider my life in an international perspective, increasing and encouraging my desire to travel and the relationship with foreign cultures. I thank hint for promoting the publication of many articles and papers relating to (his research work.
1 wish to thank doctor Davide Valenti for well-guided and deeply followed this work and the related seirul ific publications. I thank lor hiscver-rcadiness and clarity in answering all my questions. I wish to thank him for having contributed to the achievement, of a peaceful working environment, and the establishment of a true friendship.
I thank the other members of the "Group of Interdisciplinary Physics’’ and, in particular, doctor Nicola Pizzolalo for his moral and information system support, doctor Alessandro Fiasconaro. Angelo La Cognata for the frnitful discussion regarding non-Gaussian statistical distributions. Pasqualc Caldara and Stefa no Spezia.
1 thank Marcello lacono Manno for his prompt and careful technical sup-port concerning the use of t he Cometa computational grid. I wish to thank all the stafr of the Cometa computational grid for providing computational power without which the numerical simulations would not been carried out.
1 wish to thank professor Alexander Dubkov for his hospitality and helpfulness during my russian journeys, professor Nikolai Agudov. Dnhlrv and Lena Kulikov. Yuriy and Irina Ushakov, St.epan and Lusya Lebedenko for their warm welcome and care.
1 thank professor Yuri Kalmikov for his prompt report and careful comments on this thesis.
1 wish to thank Davide Gurrera for the stimulating discussion about rus-sian physical research and the Ph.D. course fellow students Rosario Grani-maul.a. Fabio Vi/.zim and Giuseppe Cannella.
Introduction
This opening chapter provides an overview of transient dynamics of Joscph-son junctions, suporconduc(ivi(> and quantum computing. Ft presents the recent studies regarding I he noise induced effects in short and long Josephson junctions which constitute the starting point of this research work
1.1 Transient Dynamics and Josephson Junctions
In lecent years a gicat attention was paid to the study of .losephson junctions because of their use both as superconducting quantum bits [1. 2. 3. 4] and nanosealo superconduct ing quantum interference devices for detecting weak inagneric Flux changes (5).
Josephson junctions represent good candidates to rcali/c superconducting quantum bits (qubits) for <|uantum information processing. For this reason, t hev wore studied at very low temperature in devices making use of charge [6]. flux |7] and phase qubits |#|. Moieovei. .Josephson junctions are widely used devi(.e foi I heir high sensitivity to magnetic flux changes For this purpose, it is employed a composition of two coupling high-Tr Josephson junctions in a superconducting ring called SQUID, the acronym for Superconducting Quantum Interference Device.
In Josephson junctions, working both at high and low temperatures, the environment affects strongly the brlmvioi of the system In high temperature superconductors (UTSs) the presence of low-frequency noise, whose intensity is related to the, fluctuations in the bias current, temperature and magnetic field, was experimentally found [9] Also in the. low temperature su-pciconduct ive devices it is veiy difficult to avoid the influence of environment ihat constitutes mainly a dccoheronce source for the system. In particular, the effects on the coherence time of weak current noise in Josephson vortex qubits (JVQs). composed by an uniform long Josephson junction, were investigated [2. 10].
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